What is the Line Parallel to y 2x 5 Passing Through the Point (4, 5)?

In this article, we'll explore how to find the equation of a line that is parallel to the given line (y 2x 5) and passes through the point ((4, 5)). We'll follow a step-by-step process to determine the equation of the line in question.

Understanding Parallel Lines

Two lines are considered parallel if they have the same slope. The slope of a line in the form (y mx b) is represented by (m). Therefore, the slope of the line (y 2x 5) is 2.

Using the Point-Slope Form

Given that the new line is parallel to the original line, it will also have a slope of 2. To find the equation of the line that passes through the point ((4, 5)), we can use the point-slope form of the equation of a line.

The point-slope form of the equation is:

(y - y_1 m(x - x_1))

Where (m) is the slope, and ((x_1, y_1)) is a point on the line.

Here, (m 2) and ((x_1, y_1) (4, 5)). Substituting these values into the point-slope form gives:

(y - 5 2(x - 4))

Simplifying the Equation

Let's simplify the equation to get it into the slope-intercept form (y mx b):

(begin{align*}y - 5 2(x - 4)y - 5 2x - 8y 2x - 8 5y 2x - 3end{align*})

The equation of the line that is parallel to (y 2x 5) and passes through the point ((4, 5)) is:

(boxed{y 2x - 3})

Verification of Parallel Lines

We can verify that the new line is indeed parallel by checking the slope. The slope of the original line (y 2x 5) is 2, which matches the slope of the new line (y 2x - 3). Since the slopes are equal, the lines are parallel and will never intersect.

Conclusion

We've demonstrated a systematic approach to find the equation of a line that is parallel to a given line and passes through a specified point. Using the point-slope form and the principle that parallel lines have equal slopes, we derived the equation (y 2x - 3).

By understanding and applying these concepts, you can solve similar problems involving parallel lines in the future.